**A Mathematical Model of
Complacency in HIV/AIDS Scenario: Sex-Structure Approach**

** **

*Department of Mathematics and
Statistics, **Federal** **University** of Technology, Minna, **Niger State**, **Nigeria**. *

E-mail: g.bolarin@futminna.edu.ng

^{* }Corresponding author: +234-80-33-88-31-37

**Abstract**

In this study we use sex-structure
approach to examine the effect of complacent sexual behaviour (risky sexual
activities) on the rate of infection of HIV/AIDS in a population. We
partitioned the population into two classes (male and female) represented by _{}to express
our model equation as a set of differential equations. We were able to express
the number of AIDS cases (male and female) as linear functions that depend on
the number of AIDS patient present in the population. We were also able to
determine the equilibra states of the model. We found that the Basic
Reproduction Number (*R _{0}*, which is the number of secondary
infections due to introduction of infective into the population) of both female
and male partitions of the population is given as

**Keywords**

Differential equation; HIV/AIDS; Sex-Structure; Complacency; Stability; Equilibrium; Basic Reproduction Number.

**Introduction**

The issue of HIV/AIDS and other Sexually Transmitted Diseases (STD) are no longer just a national or local issue but a global one.

Complacency in this study is used to mean embarking to high risk sexual behaviours such as multiple sex partners, sex with prostitutes and non-condom use or low compliance level or incorrect use of condom when it has been discovered that HIV prevalence in a community has reduced to a very low level, with the number of AIDS cases becoming less in the community. Complacency is used in the context of a community that has registered significant decreases in HIV prevalence [4].

In this study we shall consider a
sex structure model involving only male and female as sex partners i.e we shall
not consider same sex interaction because that is
not legal in most countries and we are considering sexual transmission since it
is the principal mode of infection in most countries. To model
complacency, it is assumed that behaviour change depends on the number of AIDS
patients (HIV infected persons with fully blown AIDS symptoms) in the
community. We shall consider three classes or
compartments in our model which are, susceptibles, infectives and AIDS
partitions, with population numbers in each class denoted as functions of time
by S* _{i}*(t),

**Material and Method**

* *

*Model Formulation*

For the purpose of this study we take the total population to be unity and so all the compartments sum up to 1.

We assume that, at any moment in
time, new individuals enter the heterosexually active population at a rate* _{},* a
proportion

We let * _{} *be the natural
death rate for the sexually active adults. The removal rate of susceptible
through infection is the number of new HIV infections per unit time. This rate
is important in calculating HIV incidence which by definition is the number of
new infected persons in a specified time period divided by the number of
uninfected persons that were exposed for this same time period.

The rate at
which an individual acquires new sexual partners (contact rate) per unit time
is denoted by * _{}(_{}). *Assume that a proportion of these
partners are infected male ie

Upon becoming
infected with HIV, female and male susceptibles enter the classes _{}* *and * _{} *of
infected individuals respectively. Female and male infectives
are recruited through new HIV infections described above and removed through
progression to AIDS at rate

Individual in AIDS class are
recruited through progression from the infective stage to the AIDS stage and
removed through AIDS accelerated deaths at rate _{}and natural death rate _{}and so _{} is the
average duration spent in the AIDS stage if natural deaths are assumed constant
in the model. It will be ideal to varying_{}, since there are advances in health
care there is being provided for individuals living with HIV/AIDS.

There is a constant emigration
rate *α > *0 of individuals to other countries except for the AIDS
patients. This assumption makes the model more appropriate for Nigeria where a significant proportion of the population emigrates
to developed countries for better educational facilities and in search of
employment.

*The Model Equations*

From above assumptions we have the following as our model equation.

_{}

_{}

_{ }(1)

_{}

_{}

_{}

The parameters _{} with _{} and the initial conditions for system (1) at
time t=0 are, _{} for all _{} with _{} and A_{m}(0)
= A_{m,}o > 0.

*The Disease free Equilibrium*

Following the approach of [2, 5, 6] the disease-free equilibrium is when the disease is not exiting the population and it is given as

_{}

*Endemic Equilibrium *

The endemic equilibrium of system (1) is given as

_{}

_{}

_{}

_{}

where

_{}

*The Basic Reproduction Ratio
(Number)*

The basic reproduction number _{}(the average number of secondary infection due to introduction of an
infected individual into a disease free population).

Let the probability that an infected individual in the
incubation period time t has survived to develop AIDS is _{}.

Following the approach of [7,10], if a single newly infected male is allowed into the population at equilibrium, this individual will persist and infect others with probability

_{} at time _{}

_{}

Therefore the number of female this individual will
infect over time _{} will be

_{}

Similarly if an infected female is introduced into the
population at equilibrium the number of male this individual will infect at
time _{}will
be

_{}

It is expected that the number secondary cases per generation due to an infected male is

_{}

_{}

Similarly the number secondary cases per generation due to an infected female is

_{}

_{}

Therefore the reproduction number _{} is given as;

_{ (2)}

* *

*Equilibrium in ( _{}) plane*

We will base our reasoning on the argument put forward
by Baryarama et al (see [1]). Suppose at equilibrium state in (**_{}**),
the number of female susceptible continue to increase and hence both

_{ }(3)

Similarly from sixth equation of system (1) we have

_{ }(4)

From the second equation of system (1)

_{}

Similarly from the fourth equation of system (1) we have

_{}

Now, the total population _{} (say) is given as;

_{}

Therefore, _{}, this is so because they are
proportions that can only sum to unity.

Using this fact and equations (3)-(6) we have the following result;

_{}

**Results and Discussion**

* Theorem*

Suppose from (1)
_{} Further,
suppose that _{} as an inverse, _{} Then there exist t_{A}
> 0 and ε_{A} > 0 such that for all t > t_{A}, |A_{f}(t)
- A_{f}(t_{A})| < ε_{A} and |I_{f}(t) -
I_{f}(A)| < ε_{A}. More so Sf(t) ® ¥ and t ® t_{A}.

* *

* *

*Proof*

For convenience let _{}and from (7) let _{}

So from above equation _{} we will obtain

_{}

But

_{}

If we choose _{}

Now, since _{}

then

_{}

_{ }(8)

Since

_{}

Hence the behaviour of _{} remains of little consequence to _{} since _{} Hence the
end of prove for the second part of the theorem._{} From the theorem above we can
express the equilibrium point in (**_{}**) explicitly as

More so, we can show that if _{} is linear, the
equilibrium point in (**_{}**) obtained using the theorem is
the same with the one obtained by direct method (see [1] for example).

* *

*Corollary 1*

Suppose from (1) _{} Further, suppose that _{} as an
inverse, _{}Then
there exist _{}

_{}

*Proof*

The proof follows from theorem 1.

** **

*Figure 1.** Time Trend of number of
Infected male ( _{}) and female (_{}) with Mean life time
of HIV/AIDS patient = 8 years.*

*Figure 2**. Time Trend of number of
Infected male (I _{m}) and female (I_{f}) with Mean life time of
HIV/AIDS patients = 4 years*

*Figure 3.** Time Trend of number of
Infected male (I _{m}) and female (I_{f}) with Mean life time of
HIV/AIDS patient = 2 years*

For our model we were able to show that the Basic
Reproduction Number _{} for the female and male individuals in
the population is given as _{} (from equation (2)). This shows that
the Basic Reproductive Number of female proportion in the population due to
introduction of infected male into a population is the same as the Basic
Reproductive Number of male proportion due to introduction of infected female
into a population.

This is similar to the study of Kamali et al (see [9]), in their article they found that there is a relationship between HIV prevalence and sexual behavioural change.

Also from figures 1-3 it could be noticed that the number of infected females in each case are more than that of males, this is due to the fact that the infectivity rate of male individual is higher than that of female (due to high concentration of the virus in sperm), this is a fact that has been alluded to by various authors; see [10].

**Conclusion**

The model formulated using sex-structure approach show that complacency could lead to high infection rate in a population. It can be seen from figures 1-3 that the lower the expected life time of HIV/AIDS patient in a population the higher the rate of infection, this is due to the fact that more people can embark on high sexual risk since there is an assumption that HIV/AIDS patients do not persist in a population for longer period.

From our findings it will be of great service to nations if the authorities in charge of prevention of HIV/AIDS can find a way to increase life expectancy of HIV/AIDS patients through treatment so has to prevent possible complacency behaviour by the populace.

**Acknowledgements**

I am grateful to Dr Tayo D. Olayiwola formally of Federal Medical Centre, Bida, Nigeria for his professional guidance during the conduct of this study. Also I appreciate the effort put in by the referee to review this work and for the observations and comments.

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